A Note on Goal Programming Approach to Tea Industry of Barak Valley of Assam

Application of Goal Programming in industry is an important tool to planners. A Goal programming model regarding tea production in Barak Valley[7] and its extension developed here have been presented. The proposed models yield optimal solutions. It is very easy to understand which can serve managers by providing optimal solution to different goals involving production, profit, demand etc. With the help of numerical example, the proposed models have been validated. Finally analyses of the results obtained have been carried out. It is found that total production goal can be achieved along with other goals in both the models. The two models are compared and the former is found to be more efficient than latter.


Introduction
Planning and production are known to be responsible for the development of an industry. Barak valley of Assam (India) is fu ll of g reen tea gardens and some of them are producing made tea fro m green leafs. The management of tea industries of similar nature is having different objectives to attain them within specified period of time. In this section some literatures on Goal p rogramming and its application have been presented before the analysis of the models.
Goal programming and its applicat ions in d iverse field h av e b een s tu d ied b y d ifferen t au th o rs . Co b b an d Warner [1],Trived i [2] used Goal prog ramming model to allocate resou rces in man agement related p rob lems of q u ality s e rv ice. N ja an d Ud af ia [ 3 ] ap p lied Go al programming to flour producing co mpanies. Vivekanandan etl [4] d ev elo p ed Go al P ro g rammin g mo d el fo r th e optimizat ion of cropp ing pattern fo r a particular region. Shim and Siegel [5] developed Goal programming model with sensitivity analysis to determine the decision variables and goal deviations. Pati etl [6] developed a mixed integer goal programming model for paper recycling system. The goals considered in the model have economical, social as well as quality imp licat ions on the paper recycling industry. Further we can find some of the important literature on goal p ro g rammin g in [8][9][10][11][12][13][14][15]. Recen tly Sin h a an d Sen [7] d ev elo p ed Go al p ro g rammin g mo d el reg ard in g tea production of d ifferent grades in tea industries of Barak Valley of Assam(India). For best author's knowledge, no emp irical study has been carried out so far to measure the effect of reduced cost, dual price on these developed models.
With these above motivations, an effort has been made to study the effect of reduced cost, dual price and change of priorities of goals on these models.

Mathematical Models
In [7], a Goal Programming model has been formu lated keeping in view environ mental conditions of this valley and management point of view. The authors have focused main ly on the production of different grades (according to size) of tea as the main goal and the others goals are set on profit, factory expenditure (production expenditure),demand and over use of different machines assigned for sorting and packing. These goals are given prio rit ies according to their importance. In the extended model o f [7], goals on production of different grades of teas are kept in same priority level. Both the models have been given in appendix A and B respectively.

Numerical Illustration
Now we illustrate these two models with numerical values given in appendix C. LINGO software has been used to solve these models.
The solution of model in appendix A is obtained in 10 iterations which is given below

Anal ysis of Results of First Model
The production goal and profit goal are fu lly ach ieved. The deviational variable d 3 -= 7899.008 means that there is a savings of Rs 7899.008 and the deviational variable d 4 + = 2.386518 shows that there is more demand of 2.386518 kg which can be met with the present production. Our aim is to minimize over use of both the machines but the solution reveals that the machines need not be over used as it is clear fro m the system constraints on time that the stipulated time is not fully used.
The values of the decision variables indicate that out of five grades, the production of four grades provide optimal solution.
The reduced cost of a variable is the quantity by which the solution deteriorates if we b ring that variable in the present solution with same coefficient. So, reduced cost of 2 x is 0 does not make any change in the solution if we introduce it also.
Dual p rice mean the quantity by which objective value will improve if we increase the right hand side of constraints by unit quantity. In this solution it is found that there is no dual price associated with any of the constraints.
It has also been observed that on changing priorities between (1 st and2 nd ), (1 st and 3 rd ), (1 st and 4 th ), (2 nd and 3 rd ), (2 nd and 4 th ) the present form of this model gives better solution.

Anal ysis of Results of Second Model
The second model is the extension of first model. The additional goals are set on the production of individual grades of tea. Here total production goal is achieved and moreover the deviational variable d 1 + = 1.019831 means that there is a scope of producing 1.019831kg mo re. The goals on individual production of various grades are not fully achieved in case of 2 x and 3 x wh ich are kept in second priority and also the deviational variables 21 d + = 361.6944 and d 25 + =357.0543 show towards more production of the grade 1 x and 5 x .
The values of decision variables as found, give optimal solution to this extended model. The profit goal is achieved at the expense of indiv idual production goal. One can easily understand about the reduced cost and dual price also. Change of priorities among the goals does not alter the present solution.

Concluding Remarks
This paper has main ly focused on the use of linear goal programming to make decision makers of tea industry to achieve for production management situation. The models presented in this study involve 10 constraints and 5 decision variables other than deviational variables in first model and 15 constraints in second model with same nu mber of decision variables. On comparison of these two models in this case it is found that first model gives us better objective value than the other. It is to be mention here that the second model is developed by considering the situation where tea industry may have wish to produce different grades of tea due to some co mpulsion.
The present study not only gives satisfying solution but also provides other relevant informat ion. Management can find in advance what will happen if outcome deviates from objectives. The study of range of coefficients and range of right hand side of constraints will help to understand the scenarios in better way. The present study can further be extended by introducing new constraints, variables and applying fuzzy logic to these models will be of immense help for futher study.

ACKNOWLEDGEMENTS
Author express sincere thanks to the concern tea industry for providing necessary informat ion. Author also express thanks to Prof.N.K Diwedi ,Banaras Hindu University for encouragement to carry out research in Goal Programming.